Chapter 7: Q.25 (page 361)

Show that if $X$ and $Y$ are independent, then
$E [X ∣ Y = y] = E [X] for all y$
(a) in the discrete case;
(b) in the continuous case.

Short Answer

Expert verified

The calculation for both cases is similar. Just use the fact that $X$ and $Y$ are independent.

Step by step solution

Given information(part a)

Given in the question that, $E [X ∣ Y = y] = E [X] for all y$

Explanation (part a)

Let's begin from the left side. We have that

$E (X ∣ Y = y) = \sum_{x} x P (X = x ∣ Y = y)$

$= \sum_{x} x P (X = x)$

$= E (X)$

where the second equality is the consequence of the fact that $X$ and $Y$ are independent.

Final answer(part a)

We proved that $X$ and $Y$ are independent

Given information(part b)

Given in the question that, $E [X ∣ Y = y] = E [X] for all y$

Explanation(part b)

Let's again start from the left side. For $y \in supp f_{Y}$ , we have that

$E (X ∣ Y = y) = \int_{ℝ} x f_{X ∣ Y} (x ∣ y) d x$

Because of the independence, we have that

$f_{X ∣ Y} (x ∣ y) = \frac{f (x, y)}{f_{Y} (y)} = \frac{f_{X} (x) f_{Y} (y)}{f_{Y} (y)} = f_{X} (x)$

so we have that the integral is equal to

$\int_{ℝ} x f_{X ∣ Y} (x ∣ y) d x = \int_{ℝ} x f_{X} (x) d x = E (X)$

so we have demonstrated the asserted in the two cases.

Final answer(part b)

We proved that $X$ and $Y$ are independent

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Show that if $X$ and $Y$ are independent, then
$E [X ∣ Y = y] = E [X] for all y$
(a) in the discrete case;
(b) in the continuous case.

Short Answer

Step by step solution

Given information(part a)

Explanation (part a)

Final answer(part a)

Given information(part b)

Explanation(part b)

Final answer(part b)

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Show that if Xand Yare independent, thenE[X∣Y=y]=E[X]for ally(a) in the discrete case;(b) in the continuous case.

Short Answer

Step by step solution

Given information(part a)

Explanation (part a)

Final answer(part a)

Given information(part b)

Explanation(part b)

Final answer(part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Show that if $X$ and $Y$ are independent, then
$E [X ∣ Y = y] = E [X] for all y$
(a) in the discrete case;
(b) in the continuous case.